The combination of equations (REF ) and (REF ), along with the TBA equation, gives a system of integral equations where time \(t\)  enters explicitly. Thus, it is not necessary to solve the hydrodynamic equation with finite-element analysis on the time coordinate; time can be set directly to the desired value, and the system of equations solved by recursion. The function \(u(\eta ;,x,t)\)  has the interpretation as the coordinate in real space such that the characteristic curve for the fluid mode \(\eta \)  passes by both \((u(\eta ;,x,t),0)\)  and \((x,t)\) . See [1]}.
